Math symbols in math operators

* obtains a special mathcode inside the AMS math operator setup, but here you want the standard star which is still available as \ast :

\documentclass{article}

\usepackage{amsmath}
\DeclareMathOperator*{\wslima}{w^{*}-lim}
\DeclareMathOperator*{\wslimb}{w^{\ast}-lim}
\begin{document}
\[\wslima a \quad \wslimb a\]
\end{document}

Note the reason for the redefinition is to make it possible to just use w* which is the intended markup here although as you note it does not produce the same symbol (or use the same font) as a normal math ^*.

\documentclass{article}

\usepackage{amsmath}
\DeclareMathOperator*{\wslim}{w*-lim}
\DeclareMathOperator*{\wslima}{w^{*}-lim}
\DeclareMathOperator*{\wslimb}{w^{\ast}-lim}
\begin{document}
\[\wslim a \quad \wslima a \quad \wslimb a\]
\end{document}